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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 19.3 Interference Patterns

Subsubsection Key Ideas

Applying the general rules for Maximum Constructive Interference and Complete Destructive Interference from the Interference and Path Length Difference Sections allows you to determine the spacing of the interference pattern by looking at the path length difference of the rays exiting the slits. The figure below gives a magnified look at the paths traveled by the two rays.
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On the left side, there is a diagram of the souble-slit experiment, with the slits sitting on the left a distance L away from a screen. A dashed line extends from the center of the slits to the screen at an angle theta sub m and hits a point y sub m on the screen on the right. A small circle covers the double-slit experiment on the right. On the right side, there is a zoomed-in picture of the double-slit set up inside a circle. Two slits sit at a distance d away from each other. A ray extends from each slit to the right side of the circle at an angle theta. The rays are labelled "r 1" and "r 2". Another line extends from the top slit to the bottom ray at an angle theta, indicating the path length difference.
Figure 19.3.2. Left: A diagram of the double slit experiment. The slits are separated from the screen by a distance of \(L\text{,}\) and the angle to the mth bright spot is denoted by \(\theta_m\text{.}\) The vertical distance to the mth bright spot is \(y_m\text{.}\) Right: A zoomed-in view of the double slits. The slits are separated by a distance \(d\text{.}\) Two different rays, \(r_1\) and \(r_2\) travel towards the screen at an angle \(\theta\text{,}\) making the path length difference between the two rays equal to \(d \sin\theta\text{.}\)
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If the screen is far away from the sources, the two rays in the figure above may be treated as parallel, allowing you to calculate the path-length difference using trigonometry.
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Subsubsection Activities

Activity 19.3.1. Constructive Interference.

Combine the geometric path-length difference found above and the general rule for Maximum Constructive Interference to find an equation relating \(d\text{,}\) \(\lambda\text{,}\) and \(\theta\) for the bright spots on the screen.
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Answer.
\begin{equation*} d \sin \theta = m\lambda \end{equation*}
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Activity 19.3.2. Destructive Interference.

Combine the geometric path-length difference found above and the general rule for Complete Destructive Interference to find an equation relating \(d\text{,}\) \(\lambda\text{,}\) and \(\theta\) for the dark spots on the screen.
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Answer.
\begin{equation*} d \sin \theta = \left( m + \frac{1}{2} \right)\lambda \end{equation*}
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Activity 19.3.3. The Small Angle Approximation.

In the video, you learned about using the Small-angle Approximation for interference. Use the small-angle approximation to write an equation relating \(d\text{,}\) \(y\text{,}\) \(L\text{,}\) and \(\lambda\) for points of maximum constructive interference.
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Answer.
\begin{equation*} \sin\theta \approx \theta \approx \tan\theta = \frac{y}{L} \end{equation*}
Plugging this into the maximum constructive interference expression above gives:
\begin{equation*} d \frac{y}{L} = m\lambda \end{equation*}
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The process above will allow you to find locations where the light exhibits maximum intensity (bright spots) and zero intensity (dark spots). It is also possible to determine that the intensity at any given point on the screen under the approximations described above is
\begin{equation*} I(y) = I_0 \cos^2{\left(\frac{\pi d}{L\lambda}y\right)} \end{equation*}
where \(I_0\) is the intensity of one of the point sources.
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Activity 19.3.4. Measure the Wavelength.

You are conducting the double slit experiment and you have a light source with an unknown wavelength. The slits are separated by a distance of \(0.5 \mathrm{~mm}\) and your screen is \(3.4 \mathrm{~m}\text{.}\) You measure the third bright fringe to be at a position of \(4.7 \mathrm{~mm}\) to the right of the center line on the screen. What is the wavelength of light?
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